Perron theory for positive maps and semigroups on von Neumann algebras (Q2702377)

Let \(\mathcal M\) be a von Neumann algebra, \((\mathcal H;(\cdot,\cdot))\) a Hilbert space, and \(\mathcal P\) a closed convex cone in \(\mathcal H\) such that \((\mathcal M, H, P)\) is a standard form of \(\mathcal M\). An element \(x\) in \(\mathcal H\) is positive if \(x\in \mathcal P\); a positive element is said to be strictly positive if the identity of \(\mathcal M\) is the only projection in \(\mathcal M\) whose range contains \(x\). A linear map \(A:\mathcal H\to H\) is said to be positive if \(A(\mathcal P)\subset P\). A positive map \(A\) is said to be ergodic if for all nonzero \(\xi,\eta\in \mathcal P\) there exists \(n\in \mathbb N\) such that \((\xi,A^n\eta)>0\), and \(A\) is indecomposable or irreducible if no proper closed face of \(\mathcal P\) is invariant under \(A\). This paper generalizes the Perron Theorem on the uniqueness of the eigenvector corresponding to the highest eigenvalue of a positive matrix to positive maps on \(\mathcal H\). NEWLINENEWLINENEWLINEThe main theorem states that for a symmetric, positive map \(A:\mathcal H\to H\) with \(\|A\|\) an eigenvalue, the following statements are equivalent: NEWLINENEWLINENEWLINE(i) \(\|A\|\) has multiplicity one and the associated eigenvector is strictly positive; NEWLINENEWLINENEWLINE(ii) \(A\) is indecomposable; NEWLINENEWLINENEWLINE(iii) \(A\) is ergodic. NEWLINENEWLINENEWLINEThen from it an analogous theorem about uniqueness of the ground state for a symmetric, strongly continuous positive semigroup \(\{T_t;t>0\}\) on \(H\) is deduced and is applied to prove ergodicity and indecomposability for the class of Ornstein-Uhlenbeck semigroups.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].